So guys now we're finally going to get to the last piece of information that you can derive from a proton NMR and that's going to be what we call the integration so the integration describes the relative quantities of all the different hydrogens that are present and the integration will express them as relative ratios, OK? So something that we've been talking about since the very beginning of proton NMR is that some of your peaks are going to be taller and some of them are going to be shorter now what's deceiving is to think that your integration depends only on height, it doesn't it actually also depends on the width of the peak, OK? As you can see I have an example NMR here with 3 different peaks of different heights but they also have slightly different widths, we have to take that into account when you look at integration in fact integration is just a fancy way of saying that you're taking that area under a curve you're not just looking at the exact height of it you're looking at all the little slices underneath that curve and you're adding them up together and then when you stack them up you take that area and that's really going to be what determines the relative ratio of these hydrogens, OK? Now for those you have already taken Calculus two this sounds very familiar to you this is basically the concept of Riemann sums and doing or taking integrals but for this course we're just going to let a computer to work for us so no fancy integrals that we're going to have to figure out, the computer is going to figure out these AUCs or these integrals, right? So basically the integral of a function it's going to do that work for us and it's going to give us or speed up these different distances this red line at the top is actually what we call the integration and as you can see they have different heights, one of them appears to be yay high the next one seems to be about twice as tall as the other one and the one after that seems to about three times as tall as the other, OK? What that means is that the computer actually looked at these functions looked at this as a function and it added up all the little pieces underneath it and came up with the height of this, OK? Now the important part for us isn't to be able to do the actual integral calculus to figure out the integration, it's to compare these heights and to say well how many of this type let's say this is proton A Proton B Proton C how many of A are there? How many of B? And how many of C? And what's their ratios? In this case the ratio would be a 1:2:3 ratio based on the integrations that the computer spit out for us, OK? And that's what we mean by our relative ratio, now does that mean that there's exactly 6 protons in this molecule? Actually in this case it does mean that because it looks like I have 1 3 6 protons total so in this case the ratio actually added up to the total number of protons but just you know this ratio could also work for 12 protons or 18 protons or any multiple of 6 would have basically tell us is that hey you might have 18 protons but they are in a ratio of 1:2:3 and then that would basically tell you that you've got this many of type A this many of type B and this many of type C, alright? So that said we basically have all the information that we need now to draw our own NMR spectrum I know you're getting really excited at this point so what I'm going to do is introduce this you know cumulative practice problem and then I'll have you guys try to ourselves so I want you to draw the entire NMR spectrum for this molecule, now what you're going to notice is that I did this is actually an easier way to draw it than just giving you a blank chart because I'm kind of giving you boxes to fill out as long as you have those 9 boxes filled out you get it right, OK? So let me just show you what I'm looking for first of all those that we have this PPM line right here that means this is 0 and that means this is some high number we're not going to worry about exact numbers here we're just going to worry about the order that this should be more downfield than this, they should have higher numbers as they go along, OK? So what I want you to do is that the type of proton Ha Hb Hc goes here, OK? So you're going to order those protons in order of chemical shift so this is going to be based on basically the chemical shift if it's very downfield than that should be the one that goes further just off to the left, OK? Also you are going to be responsible for.... So you don't tell me the exact value of the shift you just have to put them in order in terms of the splitting you should be able to draw the types of splits that you would get here and we're going to assume N+1 so assume that the N+1 rule works here now by the way notice that in this question I didn't say assume N+1, guess what? That's because it's up to you to determine that if your teacher if your professor has not made an explicit request to draw a tree diagram or to use a fancy formula then you're always going to assume N+1 it's the simplest way to do splitting so I want you to draw all the splits that would be predicted by Pascal's Triangle N+1 there and then finally the integrations, I want you to express the integrations as basically ratios or number of hydrogen so you could an example of an integration would be I'll just put it here an example would be let's say let me just give 3H, OK? 3H would tell me that I have 3 hydrogens that are of that type that are in that space now obviously we're going to go ahead and erase that because that's probably not the right answer for that box, OK? So go ahead and what I would try to do I'm trying to guide you through this so you can think of it one step at a time, I guess figure out what the chemical shifts are in terms of order of the protons so you can put that in these boxes here 1 2 3, OK? Then figure out what the splits are going to be, OK? Figure out what type of splits they'll be based on N+1 and then finally add up the number of hydrogens that would be of that type to get the final integrations and then I'll go ahead and solve the whole thing for you, OK? So now it's your turn go for it.

Alright guys, let's start off with the order of the protons and what we would see is that the most deshielded should be HC because HC is the closest to my electronegative atom and the most shielded should be HA. So, my order should have been HA here, HB here and HC here, because I'm assuming that HA is going to be the one closest to 0 and HC is going to be the one further down the field because the most deshielded, okay? That takes care of chemical shifts. Now, in terms of splitting, we have to use N plus 1. So, for HA I would say that N is equal to two, right? So, it's got two here. So, that means that it should be a triplet. So, I should draw a triplet for HA like that, okay? Now, for HB, what I see is a little bit more complicated, HB is getting split by three on one side and by one on the other, since we're assuming N plus 1 we can just add that all together and that's going to be, give it a second, that's going to be N equal to four, which means that we would get a quintet. So, remember that a quintet would be I believe the 1-4-6-4-1 pattern as predicted by Pascal's triangle. So, then we would try to draw that as best you can. So, 1-4-6-4-, okay? if you're not perfect it's not a big deal but we'll try do it as best as we can then finally HC. So, HC is being split by how many protons? Well, two on this side and two on this side. So, once again for HC N is equal to 4. So, we're going to get another quintet. So, let's do the same exact thing. So, it's not as nice, okay? So, there we go, we have our triple quintet and quintet and now all we need is integrations, okay? So, HA has how many hydrogen's that are of that type? six, this should be an integration of 6H why? Because we've got these three here, but I've also got these three here, remember, there was symmetry otherwise it would have had their own peak but they're symmetry. So, they're both the same. So, should be six hydrogens are type 6a, 6b how many type of that are there? Well, that's going to be 4H because I have two and two and then finally HC how many are there? just one H, okay? So, that is our fill out problem and now notice that by making these boxes and putting a kind of an order it made it easier to fill out but this is all the basics of drawing your own NMR spectrum. So, really from here. Now, after this practice problem I should be able to give you a blank NMR spectrum and you should be able to draw pretty much every single peak, every single signal, all the different splits, maybe even take a stab at the integration just from all this information, okay? Now, by the way, if you were to represent this as a ratio, the ratio would have been 1:4:6, okay? So, it's that easy. Alright, now if you do get a ratio that has multiples of numbers, let's say that, I'm just going to give a crazy example, let's say this molecule has been twice as big, so it was actually 2h it was actually 8h and it was actually 12 H, let's just say, this is just theoretical, okay? if that was the ratio then it could still be expressed like this, it could still be source with this because you could simplify all the numbers, you could divide them all by two and then you'd say? Well, there's more hydrogens but they're still in a 1:4:6 ratio. So, I'm just trying to show you that your ratios don't always add up the amount of protons you have, what they do tell you is the relative amount of each type of proton that you have, alright? so hope that made sense, let's go ahead and move on to the next topic.

Practice: PRACTICE: Which of the following molecules gives a 1H NMR spectrum consisting of three peaks with integral ratio of 3:1:6?